Zariski-dense deformations of standard discontinuous groups for pseudo-Riemannian homogeneous spaces
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arXiv
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| Format: | Preprint |
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2025
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| author | Kannaka, Kazuki Kobayashi, Toshiyuki |
| author_facet | Kannaka, Kazuki Kobayashi, Toshiyuki |
| contents | Let $X=G/H$ be a homogeneous space of a Lie group $G$. When the isotropy subgroup $H$ is non-compact, a discrete subgroup $Γ$ may fail to act properly discontinuously on $X$. In this article, we address the following question: in the setting where $G$ and $H$ are reductive Lie groups and $Γ\backslash X$ is a standard quotient, to what extent can one deform the discrete subgroup $Γ$ while preserving the proper discontinuity of the action on $X$?
We provide several classification results, including conditions under which local rigidity holds for compact standard quotients $Γ\backslash X$, when a standard quotient can be deformed into a non-standard quotient, a characterization of the largest Zariski-closure of discontinuous groups under small deformations, and conditions under which Zariski-dense deformations occur. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_03476 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zariski-dense deformations of standard discontinuous groups for pseudo-Riemannian homogeneous spaces Kannaka, Kazuki Kobayashi, Toshiyuki Differential Geometry Primary:57S30, 58H15. Secondary:22D50, 22E40, 22E46, 53C30, 58J50 Let $X=G/H$ be a homogeneous space of a Lie group $G$. When the isotropy subgroup $H$ is non-compact, a discrete subgroup $Γ$ may fail to act properly discontinuously on $X$. In this article, we address the following question: in the setting where $G$ and $H$ are reductive Lie groups and $Γ\backslash X$ is a standard quotient, to what extent can one deform the discrete subgroup $Γ$ while preserving the proper discontinuity of the action on $X$? We provide several classification results, including conditions under which local rigidity holds for compact standard quotients $Γ\backslash X$, when a standard quotient can be deformed into a non-standard quotient, a characterization of the largest Zariski-closure of discontinuous groups under small deformations, and conditions under which Zariski-dense deformations occur. |
| title | Zariski-dense deformations of standard discontinuous groups for pseudo-Riemannian homogeneous spaces |
| topic | Differential Geometry Primary:57S30, 58H15. Secondary:22D50, 22E40, 22E46, 53C30, 58J50 |
| url | https://arxiv.org/abs/2507.03476 |